2. Example

2.1. Effective Annual Interest Rate (EAR)

The first approach is to figure out the total annual return that semi-annual interest payments of 4% imply. This is the so-called Effective Annual Interest Rate or Effective Annual Return (EAR). 

In our case, the investment makes an interest payment of 4% after 6 months and another interest payment of 4% after 12 months. For each EUR invested, the total interest payments, therefore, are:

  • Interest payment after 6 months: 4% on EUR 1 = EUR 0.04
  • Interest payment after 1 year: 4% on EUR 1.04 = EUR 0.0416
  • Total interest payment = 0.04 + 0.0416 = EUR 0.0816.

 

Over an investment horizon of 1 year, an investment of EUR 1 therefore grows to EUR 1.0816. Put differently, the effective annual interest rate is 8.16%. More generally speaking, we can compute the effective annual interest rate (EAR) as follows:

  

\( \bf{EAR = \bigg(1+\frac{R}{m}\bigg)^m-1} \)
 

with: R = indicated interest (8% in our example) and m = the number of interest payments during an investment period (2 payments, in our example).

 

If we plug the numbers from our example into this equation, we can confirm the EAR of 8.16%:

 

\( EAR = \bigg(1+\frac{R}{m}\bigg)^m-1 = \bigg(1+\frac{0.08}{2}\bigg)^2 - 1 = 0.0816 =8.16\% \) 

  

Note that because of compounding, the EAR of 8.16% is larger than the indicated interest rate of 8%. The reason is that the second interest payment of 4% compounds the interest received in the first payment of 4% after 6 months.

 

Generally speaking, for a given indicated annual return (R), the effective annual return (EAR) increases as the number of compounding periods (m) goes up.

 

Once we know how to compute the EAR, we can go back to the investment example and figure out how much money we will have at the end of 5 years, if we invest EUR 100'000 today (C0):

 

\( \bf{FV_T = C_t \times (1+EAR_{R=0.08,m=2})^{(T-t)}} \)

\( \bf{FV_5= 100'000 \times 1.0816^5 = EUR \ 148'024} \)

 

With semi-annual compounding, the investment in question grows to approximately EUR 148'000.

   

Example 1:

You can choose between the following investment proposals:

    1. Annual interest rate of 10.0%, payable in annual interest payments
    2. Annual interest rate of 9.8%, payable in quarterly interest payments
    3. Annual interest rate of 9.6%, payable in monthly interest payments
    4. Annual interest rate of 9.5%, payable in daily interest payments (assuming a calendar year has 360 days).

 

Which of these investment proposals promises the highest effective annual return?

 

To find out, we can compute the EARs of the 4 proposals:

 

 1: \( EAR_{R=0.1,\ m=1} = \bigg(1+\frac{0.1}{1}\bigg)^1-1=0.1=10.00\% \)

 2: \( EAR_{R=0.098,\ m=4} = \bigg(1+\frac{0.098}{4}\bigg)^4-1=0.117=10.17\% \)

 3: \( EAR_{R=0.096,\ m=12} = \bigg(1+\frac{0.096}{12}\bigg)^{12}-1=0.134=10.34\% \)

 4: \( EAR_{R=0.095,\ m=360} = \bigg(1+\frac{0.095}{360}\bigg)^{360}-1=0.0996=9.96\% \)

 

Proposal 3 with monthly compounding therefore has the highest effective annual interest rate. We should opt for that proposal to invest our capital (assuming all proposals have identical risk and investment horizon).

 

Continuous Compounding:

What if the investment in question does not make discrete periodical interest payments such as once per quarter, month, day, hour, minute, or second but pays interest continuously? Put differently, what happens to the equation for EAR discussed before as the compounding frequency m increases to infinity?

  

It can be shown that for assets with continuous interest payments, the effective annual rate is:

  

\( \bf{EAR_{R, \ continuous}=e^R-1} \),

  

where e is the so-called Euler's number, a mathematical constant with a value approximately equal to 2.71828.

 

Example 2:

Suppose you have an investment that pays an annual interest of 10% with continuous compounding. What is the investment's effective annual rate, EAR?

 

Using the equation above, we find an EAR of 10.517%:

  

\( EAR_{0.1, \ continuous}=e^R-1 = 2.71828^{0.1} -1 = 0.10517 = 10.517\%\).

 

Summary

To compute the Future Value of an investment (C0) that makes more than one interest payment per investment period, we can proceed as follows:
 

  1. First, compute the effective annual interest rate (EAR) that is implied by the indicated annual interest rate (R) and the compounding frequency (m).

    • with discrete compounding, the EAR equals: \( EAR = \bigg(1+\frac{R}{m}\bigg)^m-1 \)
    • with continuous compounding, the EAR is \(EAR = e^R-1 \)
       
  2. Once we know the EAR, we can use the standard formula to compute Future Values, namely:
     
    \( FV_T = C_t \times (1+EAR)^{(T-t)} \).
     

When we combine the two steps (i.e., plug the relevant equation from step 1 into step 2), we get the following expressions.

 

  • For discrete compounding:  \( \bf{FV_T = C_t \times \bigg(1+\frac{R}{m}\bigg)^{m(T-t)}} \)
     
  • For continuous compounding: \( \bf{FV_T = C_t \times e^{R(T-t)}} \)
 

 

Example 3:

You can invest GBP 50'000 at an annual rate of 12% with quarterly compounding or at an annual rate of 11% with continuous compounding. In both cases, the investment horizon is 3 years. Which investment proposals generates a higher future value at the end of year 3?

 

The future value of the first investment proposal (quarterly compounding) is GBP 71'288:

 

\( FV_3 = C_t \times \bigg(1+\frac{R}{m}\bigg)^{m(T-t)} \) \(= 50'000 \times \bigg(1+ \frac{0.12}{4}\bigg)^{4 \times 3} \) \( = GBP \ 71'288 \)

 

In contrast, the future value of the second proposal (continuous compounding) is GBP 69'548:
  

\( FV_3 = C_t \times e^{R(T-t)} \) \( = 50'000 \times e^{0.11 \times 3} \) \( = GBP \ 69'548 \)

 

An investor would therefore be better off opting for the first proposal with quarterly compounding.